Navier–Stokes equations

The Navier-Stokes equations, named after Claude-Louis Navier and George Gabriel Stokes, are a set of equations that describe the motion of fluid substances such as liquids and gases. These equations establish that changes in momentum (acceleration) of fluid particles are simply the product of changes in pressure and dissipative viscous forces (similar to friction) acting inside the fluid. These viscous forces originate in molecular interactions and dictate how sticky (viscous) a fluid is. Thus, the Navier-Stokes equations are a dynamical statement of the balance of forces acting at any given region of the fluid.

They are one of the most useful sets of equations because they describe the physics of a large number of phenomena of academic and economic interest. They are used to model weather, ocean currents, water flow in a pipe, motion of stars inside a galaxy, and flow around an airfoil (wing). They are also used in the design of aircraft and cars, the study of blood flow, the design of power stations, the analysis of the effects of pollution, etc. Coupled with Maxwell's equations they can be used to model and study magnetohydrodynamics.

A variety of computer programmes (both commercial and academic) have been developed to solve the Navier-Stokes equations using various numerical methods. This approach is collectively called Computational Fluid Dynamics or CFD.

The Navier-Stokes equations are differential equations which describe the motion of a fluid. These equations, unlike algebraic equations, do not seek to establish a relation among the variables of interest (e.g. velocity and pressure), rather they establish relations among the rates of change or fluxes of these quantities. In mathematical terms these rates correspond to their derivatives. Thus, the Navier-Stokes equations for the most simple case of an ideal fluid with zero viscosity states that acceleration (the rate of change of velocity) is proportional to the derivative of internal pressure.

This means that solutions of the Navier-Stokes equations for a given physical problem must be sought with the help of calculus. In practical terms only the simplest cases can be solved in this way and their exact solution is known. These cases often involve non turbulent flow in steady state (flow does not change with time) in which the viscosity of the fluid is large or its velocity is small (small Reynolds number).

For more complex situations, such as global weather systems like El Niño or lift in a wing, solutions of the Navier-Stokes equations must be found with the help of computers. This is a field of sciences by its own called computational fluid dynamics.

Even though turbulence is an everyday experience, it is extremely difficult to find solutions for this class of problems. A $1,000,000 prize was offered in May 2000 by the Clay Mathematics Institute to whoever makes substantial progress toward a mathematical theory which will help in the understanding of this phenomenon.

Basic assumptions

Before going into the details of the Navier-Stokes equations, it is first necessary to make several assumptions about the fluid. The first one is that the fluid is continuous. This signifies that it does not contain voids formed, for example, by bubbles of dissolved gases, or that it does not consist of an aggregate of mist-like particles. Another necessary assumption is that all the fields of interest like pressure, velocity, density, temperature, etc., are differentiable (i.e. no phase transitions).

The equations are derived from the basic principles of conservation of mass, momentum, and energy. For that matter, sometimes it is necessary to consider a finite arbitrary volume, called a control volume, over which these principles can be easily applied. This finite volume is denoted by $\Omega$ and its bounding surface $\partial \Omega$. The control volume can remain fixed in space or can move with the fluid. This leads, however, to special considerations, as we shall see next.

The substantive derivative

Main article: substantive derivative

Changes in properties of a moving fluid can be measured in two different ways. This will be illustrated using an example, namely the measurement of changes in wind velocity in the atmosphere. One can measure its changes with the help of an anemometer in a weather station or by mounting it on a weather balloon. Clearly, the anemometer in the first case is measuring the velocity of all the moving particles passing through a fixed point in space, whereas in the second case the instrument is measuring changes in velocity as it moves with the fluid. The same situation arises in the measuring of changes in density, temperature, etc. Therefore, when differentiating one must separate out these two cases. The derivative of a field with respect to fixed position in space is called the spatial or Eulerian derivative. The derivative following a moving particle is called the substantive, Lagrangian or Material derivative.

The substantive derivative is defined as the operator:

${\frac {D}{Dt}}(\cdot )\equiv {\frac {\partial (\cdot )}{\partial t}}+\mathbf {v} \cdot \nabla (\cdot )$

where $\mathbf {v}$ is the velocity of the fluid. The first term on the right-hand side of the equation is the ordinary Eulerian derivative (i.e. the derivative on a fixed reference frame) whereas the second term represents the changes brought about by the moving fluid. This effect is referred to as advection.

Conservation laws

The Navier-Stokes equations are derived from conservation principles of:

• Mass
• Energy
• Momentum
• Angular momentum

Additionally, it is necessary to assume a constitutive relation or state law for the fluid.

In its most general form, a conservation law states that the rate of change of an extensive property $L$ defined over a control volume must equal what is lost through the boundaries of the volume carried out by the moving fluid plus what is created/consumed by sources and sinks inside the control volume. This is expressed by the following integral equation:

${\frac {d}{dt}}\int _{\Omega }L\;d\Omega =-\int _{\partial \Omega }L\mathbf {v\cdot n} d\partial \Omega +\int _{\Omega }Qd\Omega$

Where v is the velocity of the fluid and $Q$ represents the sources and sinks in the fluid.

If the control volume is fixed in space then this integral equation can be expressed as

${\frac {d}{dt}}\int _{\Omega }Ld\Omega =-\int _{\Omega }\nabla \cdot (L\mathbf {v} )d\Omega +\int _{\Omega }Qd\Omega$

Note that the divergence theorem was used in the derivation of this last equation in order to express the first term on the right-hand side in the interior of the control volume. Thus:

${\frac {d}{dt}}\int _{\Omega }Ld\Omega =-\int _{\Omega }(\nabla \cdot (L\mathbf {v} )-Q)d\Omega$

The expression above is valid for $\Omega$, which is a control volume that remains fixed in space. Because $\Omega$ is invariant in time, it is possible to swap the "${\frac {d}{dt}}$" and "$\int _{\Omega }^{}d\Omega$" operators. And as the expression is valid for all domains, we can additionally drop the integral.

Introducing the substantive derivative, we get when $Q=0$ (no sources or sinks):

${\frac {\partial }{\partial t}}L+\nabla \cdot \left(L\mathbf {v} \right)={\frac {D}{Dt}}L+L\left(\nabla \cdot \mathbf {v} \right)=0$

For a quantity which isn't space-dependent (so that it doesn't have to be integrated over space), D/Dt gives the right comoving time rate of change.

Equation of continuity

Conservation of mass is written:

${\frac {\partial \rho }{\partial t}}+\nabla \cdot \left(\rho \mathbf {v} \right)=0$

$={\frac {\partial \rho }{\partial t}}+\rho \nabla \cdot \mathbf {v} +\mathbf {v} \cdot \nabla \rho$

$={\frac {D\rho }{Dt}}+\rho \nabla \cdot \mathbf {v} =0$

where $\rho$ is the mass density (mass per unit volume), and v is the velocity of the fluid.

In the case of an incompressible fluid, $\rho$ does not vary along a pathline and the equation reduces to:

$\nabla \cdot \mathbf {v} =0$

Conservation of momentum

Conservation of momentum is expressed in a manner similar to the continuity equation, with vector components of the momentum replacing density, and with a “source term” to represent forces acting on the fluid. We replace $\rho$ in the continuity equation with the net momentum per unit volume along a particular direction, $\rho v_{i}$, where $v_{i}$ is the $i^{th}$ component of the velocity, i.e. the velocity along the x, y, or z direction.

${\frac {\partial }{\partial t}}\left(\rho v_{i}\right)+\nabla \cdot (\rho v_{i}\mathbf {v} )=\rho f_{i}.$

$\rho f_{i}$ is the $i^{th}$ component of the force acting on the fluid (actually the force per unit volume). Common forces encountered include gravity and pressure gradients. This can also be expressed as:

${\frac {\partial }{\partial t}}\left(\rho \mathbf {v} \right)+\nabla \cdot (\rho \mathbf {v} \otimes \mathbf {v} )=\rho \mathbf {f}$

Note that $\mathbf {v} \otimes \mathbf {v}$ is a tensor, the $\otimes$ representing the tensor product.

We can simplify it further, using the continuity equation, this becomes:

$\rho {\frac {Dv_{i}}{Dt}}=\rho f_{i}$

which is often written as

$\rho {\frac {D\mathbf {v} }{Dt}}=\rho \mathbf {f}$

In which we recognise the usual F=ma.

The equations

General form

The form of the equations

The general form of the Navier-Stokes equations for the conservation of momentum is:

$\rho {\frac {D\mathbf {v} }{Dt}}=\nabla \cdot \mathbb {P} +\rho \mathbf {f}$

where $\rho$ is the fluid density, v is the velocity vector, and f is the body force vector. The tensor $\mathbb {P}$ represents the surface forces applied on a fluid particle (the comoving stress tensor). Unless the fluid is made up of spinning degrees of freedom like vortices, $\mathbb {P}$ is a symmetric tensor. In general, we have the form:

$\mathbb {P} ={\begin{pmatrix}\sigma _{xx}&\tau _{xy}&\tau _{xz}\\\tau _{yx}&\sigma _{yy}&\tau _{yz}\\\tau _{zx}&\tau _{zy}&\sigma _{zz}\end{pmatrix}}=-{\begin{pmatrix}p&0&0\\0&p&0\\0&0&p\end{pmatrix}}+{\begin{pmatrix}\sigma _{xx}+p&\tau _{xy}&\tau _{xz}\\\tau _{yx}&\sigma _{yy}+p&\tau _{yz}\\\tau _{zx}&\tau _{zy}&\sigma _{zz}+p\end{pmatrix}}$

where the $\sigma$ are normal stresses, $\tau$ tangential stresses (shear stresses), and p is static pressure, associated with the isotropic part of the stress tensor.

The trace $\sigma _{xx}+\sigma _{yy}+\sigma _{zz}$ is always -3p by definition (unless we have bulk viscosity) regardless of whether or not the fluid is in equilibrium.

Finally, we have:

$\rho {\frac {D\mathbf {v} }{Dt}}=-\nabla p+\nabla \cdot \mathbb {T} +\rho \mathbf {f}$

where $\mathbb {T}$ is the traceless part of $\mathbb {P}$.

These equations are still incomplete. To complete them, one must make hypotheses on the form of $\mathbb {P}$, that is, one needs a constitutive law for the stress tensor as shown below.

The flow is assumed to be differentiable and continuous, allowing the conservation laws to be expressed as partial differential equations. In the case of incompressible flow (constant density), the variables to be solved for are the velocity components and the pressure. The three components of the Navier-Stokes equations plus the conservation of mass (continuity equation) conform a closed system of well-posed partial differential equations for these variables, that can be solved , in principle, for suitable boundary conditions. In the case of compressible flow the density becomes another unknown of the system, and can be determined suplementing the system with an equation of state. An equation of state usually involves the temperature of the fluid, so that the equation for conservation of energy must also be solved, coupled with the previous ones. These equations are non-linear, and analytical solutions in closed form are known only for cases with very simple boundary conditions.

The equations can be converted to Wilkinson equations for the secondary variables vorticity and stream function. Solution depends on the fluid properties (such as viscosity, specific heats, and thermal conductivity), and on the boundary conditions of the domain of study.

Special forms

Those are certain usual simplifications of the problem for which sometimes solutions are known.

Newtonian fluids

Main article: Newtonian fluid

In Newtonian fluids the following assumption holds:

$p_{ij}=-p\delta _{ij}+\mu \left({\frac {\partial v_{i}}{\partial x_{j}}}+{\frac {\partial v_{j}}{\partial x_{i}}}-{\frac {2}{3}}\delta _{ij}\nabla \cdot \mathbf {v} \right)$

where

$\mu$ is the viscosity of the fluid.

$\delta _{ij}$ is the Kronecker delta (1 for i=j; 0 for i $\neq$ j).

To see how to "derive" this, we first note that in equilibrium, p_ij=-pδ_ij. For a Newtonian fluid, the deviation of the comoving stress tensor from this equilibrium value is linear in the gradient of the velocity. It obviously can't depend upon the velocity itself because of Galilean covariance. In other words, p_ij+pδ_ij is linear in $\partial _{i}v_{j}$. The fluids that we are considering here are rotationally invariant (i.e., they are not liquid crystals). p_ij+pδ_ij decomposes into a traceless symmetric tensor and a trace. Similarly, $\partial _{i}v_{j}$ decomposes into a traceless symmetric tensor, a trace and an antisymmetric tensor. Any linear map from the latter to the former has to map the antisymmetric part to zero (Schur's lemma) and has two coefficients corresponding to the traceless symmetric part and the trace part. The traceless symmetric part of $\partial _{i}v_{j}$ is $\partial _{i}v_{j}+\partial _{j}v_{i}-{\frac {2}{d}}\delta _{ij}\partial _{k}v_{k}$ where d is the number of spatial dimensions and the trace part is $\delta _{ij}\partial _{k}v_{k}$. Therefore, the most general rotationally invariant linear map is given by

$p_{ij}+p\delta _{ij}=\mu \left(\partial _{i}v_{j}+\partial _{j}v_{i}-{\frac {2}{d}}\delta _{ij}\nabla \cdot \mathbf {v} \right)+\mu _{B}\delta _{ij}\nabla \cdot \mathbf {v}$

for some coefficients μ and μ_B. μ is called the shear viscosity and μ_B is called the bulk viscosity. It is an empirical observation that the bulk viscosity is negligible for most fluids of interest, which is why it is often dropped. This explains the factor of −2/3 appearing in this equation. This factor has to be modified in 1 or 2 spatial dimensions.

$\rho \left({\frac {\partial \mathbf {v} }{\partial t}}+(\mathbf {v} \cdot \nabla )\mathbf {v} \right)=\rho \mathbf {f} -\nabla p+\mu \left(\nabla ^{2}\mathbf {v} +{\frac {1}{3}}\nabla \left(\nabla \cdot \mathbf {v} \right)\right)$

$\rho \left({\frac {\partial v_{i}}{\partial t}}+v_{j}{\frac {\partial v_{i}}{\partial x_{j}}}\right)=\rho f_{i}-{\frac {\partial p}{\partial x_{i}}}+\mu \left({\frac {\partial ^{2}v_{i}}{\partial x_{j}\partial x_{j}}}+{\frac {1}{3}}{\frac {\partial ^{2}v_{j}}{\partial x_{i}\partial x_{j}}}\right)$

where we have used the Einstein summation convention.

When written-out in full it becomes clear how complex these equations really are (but only if we insist on writing every single component out explicitly):

Conservation of momentum:

$\rho \cdot \left({\partial u \over \partial t}+u{\partial u \over \partial x}+v{\partial u \over \partial y}+w{\partial u \over \partial z}\right)=k_{x}-{\partial p \over \partial x}+{\partial \over \partial x}\left[\mu \cdot \left(2\cdot {\partial u \over \partial x}-{\frac {2}{3}}\cdot (\nabla \cdot \mathbf {v} )\right)\right]+{\partial \over \partial y}\left[\mu \cdot \left({\partial u \over \partial y}+{\partial v \over \partial x}\right)\right]+{\partial \over \partial z}\left[\mu \cdot \left({\partial w \over \partial x}+{\partial u \over \partial z}\right)\right]$

$\rho \cdot \left({\partial v \over \partial t}+u{\partial v \over \partial x}+v{\partial v \over \partial y}+w{\partial v \over \partial z}\right)=k_{y}-{\partial p \over \partial y}+{\partial \over \partial y}\left[\mu \cdot \left(2\cdot {\partial v \over \partial y}-{\frac {2}{3}}\cdot (\nabla \cdot \mathbf {v} )\right)\right]+{\partial \over \partial z}\left[\mu \cdot \left({\partial v \over \partial z}+{\partial w \over \partial y}\right)\right]+{\partial \over \partial x}\left[\mu \cdot \left({\partial u \over \partial y}+{\partial v \over \partial x}\right)\right]$

$\rho \cdot \left({\partial w \over \partial t}+u{\partial w \over \partial x}+v{\partial w \over \partial y}+w{\partial w \over \partial z}\right)=k_{z}-{\partial p \over \partial z}+{\partial \over \partial z}\left[\mu \cdot \left(2\cdot {\partial w \over \partial z}-{\frac {2}{3}}\cdot (\nabla \cdot \mathbf {v} )\right)\right]+{\partial \over \partial x}\left[\mu \cdot \left({\partial w \over \partial x}+{\partial u \over \partial z}\right)\right]+{\partial \over \partial y}\left[\mu \cdot \left({\partial v \over \partial z}+{\partial w \over \partial y}\right)\right]$

Conservation of mass:

${\partial \rho \over \partial t}+{\partial (\rho \cdot u) \over \partial x}+{\partial (\rho \cdot v) \over \partial y}+{\partial (\rho \cdot w) \over \partial z}=0$

Since density is an unknown another equation is required.

Conservation of energy:

$\rho \left({\partial e \over \partial t}+u{\partial e \over \partial x}+v{\partial e \over \partial y}+w{\partial e \over \partial z}\right)=\left({\partial \over \partial x}\left(\lambda \cdot {\partial T \over \partial x}\right)+{\partial \over \partial y}\left(\lambda \cdot {\partial T \over \partial y}\right)+{\partial \over \partial z}\left(\lambda \cdot {\partial T \over \partial z}\right)\right)-p\cdot \left(\nabla \cdot \mathbf {v} \right)+\mathbf {k} \cdot \mathbf {v} +\rho \cdot {\dot {q}}_{s}+\mu \cdot \Phi$

Where:

$\Phi =2\cdot \left[\left({\partial u \over \partial x}\right)^{2}+\left({\partial v \over \partial y}\right)^{2}+\left({\partial w \over \partial z}\right)^{2}\right]+\left({\partial v \over \partial x}+{\partial u \over \partial y}\right)^{2}+\left({\partial w \over \partial y}+{\partial v \over \partial z}\right)^{2}+\left({\partial u \over \partial z}+{\partial w \over \partial x}\right)^{2}-{\frac {2}{3}}\cdot \left({\partial u \over \partial x}+{\partial v \over \partial y}+{\partial w \over \partial z}\right)^{2}$

$\Phi$ is sometimes referred to as "viscous dissipation". $\Phi$ can often be neglected unless dealing with extreme flows such as high supersonic and hypersonic flight (e.g., hypersonic planes and atmospheric reentry).

Assuming an ideal gas:

$e=c_{p}\cdot T-{\frac {p}{\rho }}$

The above is a system of six equations and six unknowns (u, v, w, T, e and $\rho$).

Bingham fluids

Main article: Bingham plastic

In Bingham fluids, we have something slightly different:

$\tau _{ij}=\tau _{0}+\mu {\frac {\partial v_{i}}{\partial x_{j}}},\;{\frac {\partial v_{i}}{\partial x_{j}}}>0$

Those are fluids capable of bearing some shear before they start flowing. Some common examples are toothpaste and Silly Putty.

Power-law fluid

Main article: Power-law fluid

It is an idealised fluid for which the shear stress, $\tau$, is given by

$\tau =K\left({\frac {\partial u}{\partial y}}\right)^{n}$

This form is useful for approximating all sorts of general fluids.

Incompressible fluids

Main article: Incompressible fluid

The Navier-Stokes equations are

$\rho {\frac {Du_{i}}{Dt}}=\rho f_{i}-{\frac {\partial p}{\partial x_{i}}}+{\frac {\partial }{\partial x_{j}}}\left[2\mu \left(e_{ij}-{\frac {\Delta \delta _{ij}}{3}}\right)\right]$

for momentum conservation and

$\nabla \cdot \mathbf {v} =0$

for conservation of mass.

where

$\rho$ is the density,

$u_{i}$ ($i=1,2,3$) the three components of velocity,

$f_{i}$ body forces (such as gravity),

$p$ the pressure,

$\mu$ the dynamic viscosity, of the fluid at a point;

$e_{ij}={\frac {1}{2}}\left({\frac {\partial u_{i}}{\partial x_{j}}}+{\frac {\partial u_{j}}{\partial x_{i}}}\right)$;

$\Delta =e_{ii}$ is the divergence,

$\delta _{ij}$ is the Kronecker delta.

If $\mu$ is uniform over the fluid, the momentum equation above simplifies to

$\rho {\frac {Du_{i}}{Dt}}=\rho f_{i}-{\frac {\partial p}{\partial x_{i}}}+\mu \left({\frac {\partial ^{2}u_{i}}{\partial x_{j}\partial x_{j}}}+{\frac {1}{3}}{\frac {\partial \Delta }{\partial x_{i}}}\right)$

(if $\mu =0$ but the fluid is compressible, the resulting equations are known as the Euler equations; there, the emphasis is on compressible flow and shock waves).

If now in addition $\rho$ is assumed to be constant we obtain the following system:

$\rho \left({\partial v_{x} \over \partial t}+v_{x}{\partial v_{x} \over \partial x}+v_{y}{\partial v_{x} \over \partial y}+v_{z}{\partial v_{x} \over \partial z}\right)=\mu \left[{\partial ^{2}v_{x} \over \partial x^{2}}+{\partial ^{2}v_{x} \over \partial y^{2}}+{\partial ^{2}v_{x} \over \partial z^{2}}\right]-{\partial p \over \partial x}+\rho g_{x}$

$\rho \left({\partial v_{y} \over \partial t}+v_{x}{\partial v_{y} \over \partial x}+v_{y}{\partial v_{y} \over \partial y}+v_{z}{\partial v_{y} \over \partial z}\right)=\mu \left[{\partial ^{2}v_{y} \over \partial x^{2}}+{\partial ^{2}v_{y} \over \partial y^{2}}+{\partial ^{2}v_{y} \over \partial z^{2}}\right]-{\partial p \over \partial y}+\rho g_{y}$

$\rho \left({\partial v_{z} \over \partial t}+v_{x}{\partial v_{z} \over \partial x}+v_{y}{\partial v_{z} \over \partial y}+v_{z}{\partial v_{z} \over \partial z}\right)=\mu \left[{\partial ^{2}v_{z} \over \partial x^{2}}+{\partial ^{2}v_{z} \over \partial y^{2}}+{\partial ^{2}v_{z} \over \partial z^{2}}\right]-{\partial p \over \partial z}+\rho g_{z}$

Continuity equation (assuming incompressibility):

${\partial v_{x} \over \partial x}+{\partial v_{y} \over \partial y}+{\partial v_{z} \over \partial z}=0$

Cylindrical Coordinates

Main article: Cylindrical Coordinates

The Navier-Stokes Continuity equation for cylindrical coordinates is:

${\partial u_{r} \over \partial r}+{u_{r} \over \ r}+{1 \over \ r}{\partial u_{\theta } \over \partial \theta }+{\partial u_{z} \over \partial z}=0$

The Navier-Stokes equations for cylindrical coordinates are (incomplete):

$\rho \left({\partial u_{r} \over \partial t}+u_{r}{\partial u_{r} \over \partial r}+{u_{\theta } \over r}{\partial u_{r} \over \partial \theta }+u_{z}{\partial u_{r} \over \partial z}-{u_{\theta }^{2} \over r}\right)$

Simplified version of the N-S equations. Adapted from Incompressible Flow, second edition by Ronald Panton

Note that the Navier-Stokes equations can only describe fluid flow approximately and that, at very small scales or under extreme conditions, real fluids made out of mixtures of discrete molecules and other material, such as suspended particles and dissolved gases, will produce different results from the continuous and homogeneous fluids modelled by the Navier-Stokes equations. Depending on the Knudsen number of the problem, statistical mechanics may be a more appropriate approach. However, the Navier-Stokes equations are useful for a wide range of practical problems, providing their limitations are borne in mind.

See also

• Reynolds transport theorem
• Reynolds number
• Mach number
• Reynolds-averaged Navier-Stokes equations
• Multiphase flow
• Adhémar Jean Claude Barré de Saint-Venant
• Millennium prize problem details

References

• Inge L. Rhyming Dynamique des fluides, 1991 PPUR
• A.D. Polyanin, A.M. Kutepov, A.V. Vyazmin, and D.A. Kazenin, Hydrodynamics, Mass and Heat Transfer in Chemical Engineering, Taylor & Francis, London, 2002. ISBN 0-415-27237-8

External links

• Derivation and detail discussion of Navier-Stokes equation
• A good derivation of the Navier-Stokes equations
• Derivation of the Navier-Stokes equations
• NASA page on the Navier-Stokes equations
• QEDen Millennium Prize Problems Wiki
• ADFC Navier-Stokes solver (GNU code)